LIFTED ROOT NUMBER CONJECTURE AND IWASAWA THEORY
11
4- £{VP) is cohomologically trivial, unless I 2 and p is infinite with \GP\ 2.
5. C{CK) = G(Kah'l/K) with Kah'1 the maximal abelian l-extension of K.
We need only verify 3.-5. of Lemma 2.1. For 3. observe that
X'/lnX'
-
X/lnX

X"/lnX"
is exact for all n on account of X" having no non-trivial /-torsion. Passing to the limit will
not violate exactness since the natural maps ( mod / n ) —• ( mod lm) attached to n m are
surjective (see e.g. [Har, 9.1.1, p.192]). For 4., see [GW1, p.282] when p is finite, and note that
that assertion is trivial when Gp = 1. Finally, 5. results from the short exact sequence CK -+
CK
G{K^h/K)
of class field theory, which displays the Galois group of the maximal abelian
extension Kab of K as the quotient of the idele class group by its connected component CK at
1. Since C°K is a divisible group (see [AT, p.37]), CK / C% ^ G(K&h/K) / (G(Kah/K))in, and
this implies 5., because
G(Kah,l/K)
is the maximal pro-/ factor group of
G(K*h/K).
This
finishes the proof of the lemma.
Note that, by 1. in Lemma 2.1, £(X) has finite projective dimension over Z/G, whenever X
is finitely generated with finite projective dimension over 7LG.
As a consequence of the lemma, (D2.1) when /-completed looks like
Zt®E - Zi (g) A ZiSL
(D2.1/) £{JS) - £(V*) - Z
Z
®A *
£(CK) - £(V) - AtG
with exact columns and rows as shown.
L E M MA 2.2. 1. 7L\®E^ £{Js) is infective.
2. £(V3) is cohomologically trivial, unless I = 2 and some real prime of k becomes
complex in K.
In order to prove 1., we start out from an element
(..., en+i mod Ein+1, en mod E1",...) G £(E) =11®E
that is sent to 1 in £(Js)- Then e
n
+i G E is an /
n + 1
t h power in J5, whence an /
n + 1
t h power
locally everywhere. By [AT, p.96], en+\ = a^
+ 1
with some an+\ G Kx, so in fact a
n
+i G E.
Since
en+\Eln
maps to
enEln,
it follows
enEln

E1™.
Claim 2. of Lemma 2.2 is a consequence of (D2.1/) together with the just observed injectivity
of Zt (8) E -+ £(Js)- Indeed, now also Z{ ® A - £(V*) is injective and so £(^J) ~ £(V*)/£{A)
is cohomologically trivial by 4. of Lemma 2.1. The proof of the lemma is complete.
R E M A R K S .
1. From now on a citation (D2.1/) refers to (D2.1/) with the injectivity of the two upper
left vertical maps filled in.
2. The translation functor applied to the image of the fundamental class under H2(G, CK) —*
H2(G,G(K^l/K)) yields our sequence G(K^l/K) ~ £(2J) AtG, as follows from
the Safarevic-Weil theorem [Ta3, p. 199].
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